$S(0)=150, u=1.2, d=.8, X(\text{strike price})=120, r=10\%$, expiration $T=2$ where $S(t)$ is price of stock at time $t$, $S_u$ is price of stock if it goes up at time $1$ and $S_{uu}$ is price of stock if it goes up again at time $2$, similar for $S_d, S_{dd}, S_{ud}, S_{du}$
From this I got the value of the call option at time $0$ equal to $C(0)=55.79$ using black-scholes formula. Then I got the payoff at time $1$ as $100.91$ if stock goes up and $21.82$ if stock goes down.
Then I had to find the price of the call option at time $1$ for both scenarios (stock goes up or down) and I got $C(1)=100.91$ if stock goes up and $C(1)=21.82$ if stock goes down.
Is the payoff at time $1$ supposed to be equal to the value of the call option at time $1$ if this is a 2 period European call? If yes can anyone explain why they are equal?
EDIT
This is what we derived in class for the value of the call option with expiration $T=2$
$$C(0)=\frac{1}{(1+r)^2}(P_*^2C_{uu}+2P_*Q_*C_{ud}+Q_*^2C_{dd})$$
where $P_*=\frac{1+r-d}{u-d}$ and $Q_*=\frac{u-1-r}{u-d}$
$S_u=210, S_{uu}=294, S_{ud}=168, S_d=120, S_{dd}=96$
$C_{ab}$ is the payoff at $ab$, $C_{uu}=174, C_{ud}=48, C_{dd}=0, C_u=100.91, C_d=21.82$
Then I got $C(1)=100.91$ if stock goes up at time $1$ and $C(1)=21.82$ if stock goes down at time $1$.
I got $C_u$ with this $C_u=\frac{1}{(1+r)}(P_*C_{uu}+Q_*C_{ud})$ and $C_d$ with $C_u=\frac{1}{(1+r)}(P_*C_{du}+Q_*C_{dd})$
You might know how to do this a different way, do you know if the payoff should equal the price of the option at time $1$ if this is a $2$ period European call?
How did you obtain the Black-Scholes price if you don't have the volatility $\sigma$? This exercise clearly suggest a binomial tree approach, which is a very simplistic way to price calls and puts. The aim is to find the option price at $t=0$. First, you need to describe all possible stock price movements, which is quite easy since the stock can only go up or down.Note that you have the following possibilities: (1) up,up (2) up down (3) down up (4) down down.
When you have determined the possible stock prices at maturity $T = 2$, you can immediately find the option value since this is the pay off, that is, for a call option: $\max(S_{2}-X,0)$ (where $S_2$ is the stock price at maturity). Once you have these pay-offs you can use the binomial tree formula to find the option price at $t=1$ (note that this is not the pay-off at $t=1$) and accordingly at $t=0$.